Page:A Treatise on Electricity and Magnetism - Volume 1.djvu/409

317.] The current at any point of the first medium is the same as would have been produced by the source $$S$$ together with a source $$\fracS $$ placed at $$I$$ if the first medium had been infinite, and the current at any point of the second medium is the same as would have been produced by a source $$\frac $$ placed at $$S$$ if the second medium had been infinite.

We have thus a complete theory of electrical images in the case of two media separated by a plane boundary. Whatever be the nature of the electromotive forces in the first medium, the potential they produce in the first medium may be found by combining their direct effect with the effect of their image.

If we suppose the second medium a perfect conductor, then $$k_2 = 0,$$ and the image at $$I$$ is equal and opposite to the course at $$S.$$ This is the case of electric images, as in Thomson's theory in electrostatics.

If we suppose the second medium a perfect insulator, then $$ k_2= \infty,$$ and the image at $$I$$ is equal to the source at $$S$$ and of the same sign. This is the case of images in hydrokinetics when the fluid is bounded by a rigid plane surface.

316.] The method of inversion, which is of so much use in electrostatics when the bounding surface is supposed to be that of a perfect conductor, is not applicable to the more general case of the surface separating two conductors of unequal electric resistance. The method of inversion in two dimensions is, however, applicable, as well as the more general method of transformation in two dimensions given in Art. 190.

Conduction through a Plate separating Two Media.

317.] Let us next consider the effect of a plate of thickness $$AB$$ of a medium whose resistance is $$k_2,$$ and separating two media whose resistances are $$k_1$$ and $$k_2,$$ in altering the potential due to a source $$S$$ in the first medium.

The potential will be